Yesterday evening, the Senate Committee on Homeland Security and Governmental Affairs approved a bill that would allow the president to "shut down private sector or government networks" in the event of a cyber attack that threatened lives or other "major damage."
From The Hill:
"It's been frustrating to read some of the misrepresentations of our bill in the cybersphere," [Susan Collins, Republican Rep from Maine and co-sponsor of the bill] said, arguing the new bill actually circumscribes the president's existing authority and puts controls on its use. "I believe the substitute amendment we’re offering strengthens those protections even more."
Maybe that is because the objection is not against allowing the President to shut off the internet in certain circumstances, but allowing the President to shut off the internet at all.